© Strojni{ki vestnik 47 (2001 )1,4-14 ISSN 0039-2480 UDK 624.97:621.396:539.3 Izvirni znanstveni ~lanek (1.01) © Journal of Mechanical Engineering 47 (2001 )1,4-14 ISSN 0039-2480 UDC 624.97:621.396:539.3 Original scientific paper (1.01) Stabilnost antenskega stebra ob upo{tevanju hkratnega delovanja lastne te`e in zunanje obremenitve The Stability of an Antenna Column under the Simultaneous Action of its Own Weight and an Effective Load Juraj Saucha - Jerko Rado{ - ^edomir Ivakovi} Prispevek obravnava določitev elastične stabilnosti antenskega stebra ob hkratnem delovanju lastne teze in zunanje vzdolžne tlačne obremenitve. Določitev kritične kombinacije lastne teze stolpa in vzdolžne tlačne sile temelji na rešitvi vodilne diferencialne enačbe stebra v obliki Maclaurinove vrste s hitro konvergenco zaradi uporabljene zamenjave. Dobljeni rezultati s kakovostjo natančne rešitve vodilne diferencialne enačbe so primerni za oceno natančnosti približne rešitve, ki predpostavlja linearno sodelovanje teze stebra in vzdolžne tlačne sile.Ugotovili smo, da so rezultati približne rešitve znotraj dveh odstotkov. © 2001 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: stebri antenski, stabilnost, obremenitve kritične, enačbe diferencialne) This paper deals with the determination of the elastic stability of an antenna column subjected to the simultaneous actions of its own weight and an effective load as an axial compressive force. The determination of the critical combination of the weight of the column and the axial compressive force was based upon the solution to the governing differential equation of the column in the form of a Maclaurin‘s series with a rapid convergence due to the introduced substitution. The obtained results, with a quality of the exact solution to the governing differential equation, were relevant for the evaluation of the accuracy of the approximate solution which assumed the linear interaction of the weight of the column and the axial compressive force. It has been established that the approximate solution differed from the results obtained in this paper by -2 %. © 2001 Journal of Mechanical Engineering. All rights reserved. (Keywords: antenna columns, stability, critical loads, diferential equations) 0 UVOD Antenski steber smo analizirali in oblikovali kot nosilec, saj je obremenjen upogibno zaradi prečne obremenitve (vihar), in tlak zaradi lastne teže in zunanje obremenitve. V običajni praksi zapleteno analizo stebra drugega reda nadomestimo s približno, ki uporablja povečevalni količnik. Največji moment drugega reda preprosto dobimo z množenjem največjega momenta prvega reda s tem količnikom ([1] in [2]). V naslednjem približku vse druge momente prerezov vzdolž antenskega stebra dobimo z množenjem prvih momentov s povečevalnim količnikom [6]. Da določimo vrednost povečevalnega količnika, moramo najprej preučiti stabilnost stebra, saj je povečevalni količnik odvisen od razmerja dejanske vzdolžne obremenitve in kritične vzdolžne sile stebra. Določitev kritične vzdolžne sile antenskega stebra izhaja iz preučitve elastične 0 INTRODUCTION An antenna column has to be analyzed and designed as a beam column because it is subjected to both bending, due to transverse loading, and compres-sion, due to its own weight and an effective load. In practice, the complicated second-order analysis of a beam column is replaced, in an approximate sense, by the use of the magnification or amplification factor. The maximum second-order moment is obtained simply by multiplying the maximum first-order moment by this factor ([1] and [2]). In a further approximation, all the secondary moments along the antenna column can be obtained by increasing the primary moments by the amplification fac-tor [6]. To determine the value of the amplification factor the stability of the column has to be analyzed first, be-cause the amplification factor is a function of the ratio of the actual axial load to the critical axial load of the column. The determination of the critical axial load of the antenna column results from a study of the elastic grin^SfcflMISDSD VH^tTPsDDIK stran 4 J. Saucha, J. Rado{, ^. Ivakovi}: Stabilnost antenskega stebra - The Stability of Antenna Column stabilnosti nosilnega stebra z zvezno spremenljivim prerezom ob hkratnem delovanju njegove lastne teže (porazdeljena vzdolžna obremenitev) in zunanje obremenitve (vzdolžna tlačna sila, delujoča na prostem koncu). Približno vrednost kritične kombinacije teže stebra in vzdolžne tlačne sile dobimo ob predpostavki linearne povezanosti teže stebra in vzdolžne tlačne sile. To kritično kombinacijo izrazimo s kritično težo stebra in kritično vzdolžno tlačno silo, ki deluje na steber zanemarljive lastne teže [6]. Kritično vzdolžno tlačno silo nosilnega stebra z zvezno spremenljivim prerezom določimo po razcepnem postopku z reševanjem vodilne diferencialne enačbe z uporabo Besselovih funkcij ([4] in [7]). Kritično lastno težo takega stebra približno določimo (vendar z veliko stopnjo natančnosti) z reševanjem vodilne diferencialne enačbe v obliki neskončne potenčne vrste [5], ali pa z uporabo energijskega postopka po Galerkinu [6]. Približne energijske postopke lahko uporabimo za določitev kritične kombinacije lastne teže stebra in vzdolžne tlačne sile, npr. po postopku Rayleigh-Ritz [3]. Prikazana proučitev elastične stabilnosti antenskega stebra, obravnavanega kot nosilni steber z zvezno spremenljivim prerezom, izpostavljenim hkratnemu delovanju lastne teže in vzdolžne tlačne sile, uporablja razcepni postopek. Določitev kritične kombinacije teže stebra in vzdolžne tlačne sile uporablja rešitev vodilne diferencialne enačbe v obliki Maclaurinove vrste. Določitev kritične obremenitve je izvedena z največjo natančnostjo. Dobljeni rezultati s kakovostjo natančne rešitve vodilne enačbe so primerni za ovrednotenje natančnosti približne rešitve, ki predpostavlja linearno povezanost teže stebra in vzdolžne tlačne sile. To ovrednotenje je namen pričujočega proučevanja. 1 POSTAVITEV VODILNE DIFERENCIALNE ENAČBE Slika 1 prikazuje steber z zvezno spremenljivim prerezom. Vztrajnostni moment prereza stebra se vzdolž smeri x spreminja parabolično: I(x) = I2 in porazdeljena vzdolžna sila zaradi lastne teže stebra je prav tako parabolična: qG(x) = stability of a cantilevered column with a continuously changing cross-section subjected to the simultaneous action of its own weight ( a distributed axial load ) and the effective load ( an axial compressive force acting at the free end ) . The approximate value of the critical combination of the weight of the column and the axial compressive force can be obtained by assuming a linear interaction of the weight of the column and the axial compressive force. This critical combination is expressed by the critical weight of the column and the critical axial compressive force acting on the column with a negligible weight of its own [6]. The critical compressive axial force of the can-tilevered column with a continuously changing cross-section can be determined, in a bifurcation approach, by solving the governing differential equation by means of Bessel functions ([4] and [7]). The critical intrinsic weight of such a column can be determined approximately ( yet with a high degree of accuracy ) by a solution to the governing differential equation in the form of an infinite power series [5] or, using an energy approach, by Galerkin’s method [6].The approximate energy methods can be used for determing the critical combination of the intrinsic weight of the column and the axial compressive force, for instance the Rayleigh-Ritz method [3]. The presented study of the elastic stability of an antenna column, treated as a cantilevered column with a continuously changing cross-section subjected to the simultaneous action of its own weight and the axial com-pressive force, uses the bifurcation approach. The determination of the critical combination of the weight of the column and the axial compressive force is determined from the solution to the governing differential equation in the form of a Maclaurin’s series. The determination of the criti-cal load has been worked out with maximum precision. The obtained results, with a quality of the exact solution to the governing equation, are relevant for the evaluation of the accuracy of the approximate solution assuming the linear interaction of the weight of the column and the axial com-pressive force. This evaluation is the aim of this study. 1 FORMULATION OF THE GOVERNING DIFFERENTIAL EQUATION Fig. 1 shows a cantilevered column with a continuously variable cross-section. The moment of inertia of the column cross-section varies along the x axis in accordance with the parabolic law: a1 (1) + lj and the distributed axial load due to the intrinsic weight of the column in accordance with the parabolic law: — (2), | lgfinHi(s)bJ][M]lfi[j;?n 01-1_____ stran 5 I^BSSIfTMlGC J. Saucha, J. Rado{, ^. Ivakovi}: Stabilnost antenskega stebra - The Stability of Antenna Column Sl. 1. Steber z zvezno spremenljivim prerezom Fig. 1. Column of variable cross-section kjer sta n in p nespremenljiva, odvisno od geometrijske oblike stebra in njegovega prereza. Ravnovesna enačba stebra ima obliko [6]: where n and p are constants depending on the geom-etry of the column and its cross-section. The equilibrium equation of this column takes the form [6]: [EI(x)w"] + Fa (x)w + qG (x)w = 0 kjer so ()'=ddx(),()"=d d x 22 () in where ()'=ddx(),()'=dd x 2 2 () and x (3), (4). Fa(x) = F + j q G ( x ) d x 0 Z vstavitvijo (4) v (3) dobimo po integraciji Substituting (4) into (3) after integrating (in- tegracija tretjega člena po delih): cluding integrating by parts of the third term) gives: x [ EI(x)w"]" + Fw + w'J qG (x)dx + C 1 = 0 (5) 0 Ob predpostavitvi majhnih pomikov w, lahko Assuming we have small deflections w, the prečno silo FT izrazimo [1] transverse force FT can be expressed as [1]: kjer je vzdolžna sila: in strižna sila: Za x = 0, FL = -F in FT = 0, dobimo: Prav tako je: FT=FLw' + F Q where the longitudinal force is: x FL = -Fa = -F - j" q G ( x ) d x 0 and the shearing force is: Fq= dx = dx[ -EI(x)w ] = - [ EI(x ) w"] For x = 0, FL = -F and FT = 0: FQ (0)-Fw'(0) = 0 FQ (0)=[dML= d [-EI(x)w] = -E Further: dI (x) dx w"(0)-EI(0)wm(0) (6), (7) (8). (9). (10). maimskixmmm ^BsfUWHIK | stran 6 J. Saucha, J. Rado{, ^. Ivakovi}: Stabilnost antenskega stebra - The Stability of Antenna Column Upogibni moment M je nična prostem koncu stebra. Tako je: The bending moment M vanishes at the (11) in zato: Z vstavitvijo (12) v (10) dobimo: in (16) v (9) daje: Enačba (5) za x = 0 dobi obliko: Ew"(0) Upoštevajoč (14) in (12) sledi: free end of the column. Hence: M(0) = -EI(0) w"(0 ) = 0 and thus: w"(0) = 0 Substituting (12) into (10) gives: FQ(0) = -EI(0 ) w-(0 ) and subsituting (16) into (9) gives: EI(0)w"'(0) + Fw'(0) = 0 Equation (5) for x = 0 takes the form: dI (x) + EI(0)wm(0) + Fw'(0) + C1 = 0 Considering (14) and (12) , it follows that: C =0 (12) (13) (14). (15). Integral tretjega člena v enačbi (5), The integral in the third term of equation upoštevajoč (2), daje: (5), considering (2), yields: Teža stebra je: Vstavitev (1) v (16) daje: x \qG(x)dx 0 (a +l) qG2 (a + x)p+1 - a p+1 p+1 FG=\qG(x)dx 0 (a + l) p +1 The weight of the column is: qG2 (a + l)p+1 - ap+1 jqG(x)dx = FG 0 (a p p +1 substituting (17) into (16) gives: (a + x)p+1 - ap+1 p + 1 p + 1 + l)p+1 - a (16). (17) (18). 0 Če (1) in (18) vstavimo v (5), dobimo po If (1) and (18) are substituted into (5), it deljenju z EI2/(a + l)n follows after dividing by EI2/(a + l)n that: [(a+x)w"]+ F (a+l) n w-+ FG (a+l) n (a+l) -a =0 (19). Če uvedemo konstanti k in k, dobimo: F k1 =G (a + l)n EI 2 If constants k1 and k2 are introduced as follows: 1 (20), (a + l)p+1 - ap+1 F EI (a + l ) n 2 Vstavitev k1 in k2 v (19) daje: substitution of k1 and k2 into (19) gives: [(a + x)n w"]' + { k1 [(a + x)p+1 - ap+1 ] + k2 } w = 0 (21). (22). 2 REŠITEV VODILNE DFERENCIALNE ENAČBE Natančna rešitev enačbe (22) ne obstaja. Rešitev v obliki Maclaurinove vrste tudi ni ustrezna. 2 SOLUTION TO THE GOVERNING DIFFERENTIAL EQUATION There is not an exact solution to equation (22 ).The solution in the form of a Maclaurin’s series has not been isfFIsJBJbJJIMlSlCšD I stran 7 glTMDDC J. Saucha, J. Rado{, ^. Ivakovi}: Stabilnost antenskega stebra - The Stability of Antenna Column Konvergenca vrste je odvisna od zveze I2/I1 < 2n, kar satisfactory. The convergence of the series was condi- omejuje uporabnost rešitve na stebre z majhno tioned by the relation I2/I1 < 2n, which limited the applica- koničnostjo. Poleg tega je konvergenca zelo počasna, tion of the solution to the columns with small conicity. In kar zahteva izračunavanje velikega števila členov addition, the convergence was very slow, which meant vrste. that a lot of terms in the series had to be calculated. Zato smo, da bi dobili vrsto s hitro Therefore, with the aim of obtaining a series konvergenco, tudi za velika razmerja I2/I1, uporabili with a rapid convergence, which exists even for the nadomestitev: big ratia I2/I1, a substitution t was introduced: t = -b-ln 1 kjer je b ustrezna konstanta. Če zapišemo: Here b is an arbitrary constant. If it is written: dw d2w = ,d3w „ dt =v, dt 2 V' dt 3 = v dobi enačba (22) obliko: equation (22) takes the form: n-3 n-2 v+ K (n-2) v+K ebv=0 kjer sta (n-p-3) (n-2) I J 2 where: K =k 11 2 b -(n-2) K =k 22 2 b (23), (24), (25), (26), (27). Rešitev enačbe (25) lahko iščemo v obliko The solution to equation (25) can be sought Maclaurinove vrste: in the form of a Maclaurin’s series: t . t -v'0+-1! 2! (r ) (28). Odvode v0r) = v(r) lahko določimo po The derivatives v0r) = v(r) can be de- naslednjem postopku. Iz robnih pogojev w’ = C za termined as follows. From the boundary condition x = 0 (strmina elastične krivulje na prostem koncu w’ = C2 for x = 0 (the slope of the elastic curve at the stebra) sledi ob upoštevnaju t(x = 0) =-b-1n(1 + 0) = free end of the column), it follows, considering 0 po (23): t(x = 0) =-b.1n(1 + 0) = 0 by (23): dw1 ~b dx J 0 eb =C in zato: and hence: v0 =--C2=C b (29). Iz robnega pogoja w"(x = 0) = 0, zaradi From the boundary condition w "(x = 0) = 0, M(x = 0) = 0 (upogibni moment na prostem koncu because M(x = 0) = 0 (the bending moment at the free stebra), sledi ob upoštevanju t(x = 0) = 0: end of the column) , it follows, considering t(x = 0) = 0: d2w dx2 b 2 2 0 b 2 0 — v'0eb+—v0eb =0 a a in zato: and hence: v;=--C b (30). Iz (25), upoštevajoč t(x = 0) = 0 ter (29) in (30), izhaja: From (25), considering t(x = 0) = 0 and (29) and (30), it follows that: v0= b C (31). Zaporedno odvajanje v", izraženo iz (25) za Successive derivating of v " expresed from grin^SfcflMISDSD ^BSfiTTMlliC I stran 8 J. Saucha, J. Rado{, ^. Ivakovi}: Stabilnost antenskega stebra - The Stability of Antenna Column t = 0, upoštevajoč (29), (30) in (31) daje: (25) for t = 0 considering (29),(30) and (31), produces: v0, v0I V,.............., vo(r ) , ..... Nadalje uporabimo robni pogoj w'(x = 1) = 0. Further, the boundary condition w (x = 1) = Zaradi (23): 0 has to be used. According to (23): t(x = l) = t1=-b-l n\— (32). Iz (1), za x = 0 je: Following (1), for x = 0: a + l (I2 1 kar vstavljeno v (32) daje: which substituted into (32) gives: (33), t=-b-ln I2 (34). Robni pogoj w '(x = l) = 0 daje: The boundary condition w (x = l) = 0 gives: fdw\ fdw\ (dt\ {dx)x=l {dt)t=t1 {dx)x=l (35) ali, upoštevajoč (24) in (23), or, considering (24) and (23): in zato: and hence: (36) (v)t=t1=0 (37). Če (37) izrazimo z (28), dobimo: If (37) is expressed by (28), it follows that: 1! 2! r! (38). (r) Izraz (38) vodi k enačbi P(K ,K2) = 0 (pomeni The expression (38) leads to the equation polinom), če količniki, s katerimi množimo odvode P(K ,K) = 0, (P denotes polynomial) , if the coeffi- cients multiplying the derivatives v0(r) in (38) form a converging series. The derivative v0(r), obtained if v" expressed from (25) is (r - 2) - times consecutively derivated, takes the form: v (38), dajejo konvergentno vrsto. Odvod v0), dobljen kot v " iz (25), odvajamo (r - 2) - krat zaporedno, dobi obliko: n-3 n-2 v(r) = v(r-1) + v(r -2) -K (r-2) (r-2) n-p-3 (r-3) (r-2)(r-3)...(r-i) fn-p-3 i [r-(i+1)] (i-1)! "' ' v +... (r-2)(r-3)...[r-(r-1)] fn-p-3 +-----------(r - 2)!------------' v + (39). v(r-2) + (r-2) ^ n-2 v 1! b 0 + (r-2)(r-3)...(r-i) _ r n - 2 j v.HM)] + ... + (i-1)! l b (r-2)(r-3)...[r-(r-1)] (n-2^2 ' (r-2)! Analiza izraza (39) kaže, da lahko (38) The analysis of the expression (39) shows zapišemo v obliki: that (38) can be written in the form: isfFIsJBJbJJIMlSlCšD I stran 9 glTMDDC J. Saucha, J. Rado{, ^. Ivakovi}: Stabilnost antenskega stebra - The Stability of Antenna Column kjer je cr,i količnik, ki odvod v(r) v (38) množi z zmnožkom (K1Ks2 )i . (Tu je i povezan z določeno dvojico vrednosti eksponentov q in s). Raziskava konvergence vrste: EEcr,i t1r (K1q K2s )i =0 (40), where cr ,i is the coefficient which in the derivative v0(r) , appearing in(38), multiplies the product (K1q K2s )i . (Here i is connected with a particular couple of the values of the exponents q and s.) The investigation of the convergence of the series: 00 Tcr r=0 t1r r! (41) je pokazala, da je to hitro konvergentna vrsta, katere konvergentnost je hitrejša, če je I2/I1 manjši ter konvergenca obstaja za vsako razmerje I2/I1 in zato lahko izberemo b = 1. Upoštevajoč hitro konvergenco (41), lahko vsoto C aproksimiramo kot: showed that this was a rapidly converging series, with a convergence more rapid as I2/I1 becomes smaller and that a rapid convergence existed for any ratio I2/I1. The convergence does not depend upon the arbitrary con-stant b introduced in (23) and thus its value can be chosen as b = 1. Considering the rapid convergence of (41) its sum Ci can be approximated as follows: Ci =2cr ,i (42), r! kjer je R dovolj visoka stopnja najvišjega odvoda , izračunanega v (39). Prav tako se je izkazalo, če (R) je R dovolj visoka, C množi zmnožek ( kqKs2 ) eksponentoma q in s, proti največjim vrednostim v odvodih v0() do v0R), praktično izgine. Zaradi tega, ob zadosti visokem R, vrste (40) praktično preide v vrsto: W, f R t1r i=0 V_r=0 r ki ima, zaradi končnega števila členov, obliko polinoma P(K ,K) z vrednostjo nič. Oznaka i aks v enačbi (43) pomeni dvojico eksponentov q in s v kombinaciji z največjimi vrednostmi v odvodih v0r) do v(R). Enačba P(K ,K) = 0 omogoča določitev kritične kombinacije podkritične teže FG stebra in tlačne sile F’ ki povzroči uklon sttebra določene geometrijske oblike. Zaradi (26), (27), (20), (21) in (1), upoštevajoč b = 1, dobimo: where R is a sufficiently high order of the highest derivative v0(R) calculated in (39). Further, it is clear that if R is sufficiently high, Ci multiplying products (K1qK2s)i with exponents q and s tending to the maximum values appearing in derivatives v0(r) ending by v0( R) , practi-cally vanishes. Considering this, with a sufficiently high R, series (40) is practically equal to the series: (k 1 q k 2s ) i (43), which, due to a finite number of terms, has the form of a polynomial P(K1,K2) equal to zero. The mark imax appearing in (43) denotes a couple of exponents q and s in a combination of their maximum values ap-pearing in derivatives v0(r) ending by v0(R) . The equation P(K1,K2) = 0 makes it possible to determine the critical combination of the subcriti-cal weight FG of the column and the compressive force F’cr which will cause buckling of a column with a particular geometry. According to (26),(27),(20),(21) and (1), considering b = 1 it follows that: K EI2 I1 p + 1 I2 -1 I K = F I2 a 2 EI2' I1 (44), (45). Veličino a2 lahko spremenimo, upoštevaje (1) v: a2 = a-2¦l2 Vstavitev (46) v (44) in (45) daje: The quantity a2 can be transformed, considering (1): (46). I2 Substituting (46) into (44) and (45) gives: FGl K EI wirmsksrnm^ (47) VH^tTPsDDIK stran 10 J. Saucha, J. Rado{, ^. Ivakovi}: Stabilnost antenskega stebra - The Stability of Antenna Column K2 EI Fl2 2 I2 I1 Videli smo, da K1 in K vsebujeta FG, F, I/I in eksponenta n in p, ter zato enačba P(K,K2) = 0 kakor je bilo povedano, omogoča določitev kombinacije FG in F’ , ki povzroča uklon stebra znane geometrijske oblike. cr Določitev kritične kombinacije (F + F) je bila izvedena kakor sledi. Najprej za F = 0 je enačba P(K,K2) = 0 spremenjena v P(K1) = 0. Iz najmanjšega korena K te enačbe, je določena kritična teža FGc stebra, brez dodatne vzdolžne tlačne sile F, z uporabo enačbe (47). Nato je za različna razmerja FG/FGc , ki določajo vrednost F in tako tudi vrednost K1, enačba P(K1, K) = 0 spremenjena v posebno enačbo P(K2) = 0. Iz najmanjšega korena K teh enačb, so bile kritične sile F’ , ki skupaj z dano FG povzročajo uklon, določene z enačbo (48). Na ta način, za F = 0, je bila določena F r0 kot kritična tlačna sila, ki deluje v stebru zanemarljive teže. Vrednosti FG , F 0 in F’ so bile ovrednotene za naslednje stebre z geometrijskimi oblikami, ki jih določata n in p: 1. n = 4, p = 2 (steber v obliki prisekanega stožca ali piramide oz. votlega prisekanega stožca ali piramide); 2. n = 3, p = 1 (to se nanaša na steber nespremenljive debeline in linearno odvisne širine prereza oziroma približno na steber v obliki stožčaste cevi nespremenljive debeline stene); 3. n = 2, p = 0 (to se približno nanaša na steber, zgrajen iz štirih palic nespremenljivega prereza, postavljenih vzdolž robov navidezne prisekane piramide, združenih z mrežno polnitvijo zanemarljive teže). Veličina d je narisana v odvisnosti od razmerja I2/I1 (sl.1) ter za omenjene vrednosti n in p v diagramu 1. Kritična lastna teža stebra FGc je povezana z dG kot: (48). We can see that K1 and K2 contain FG, F, I2/I1 and exponents n and p , and thus the equation P(K1,K2) = 0, as was mentioned above, makes possible the determination of the combination of FG and F’cr causing the buckling of a column of a particular geometry. The determination of the critical combination (FG + F)cr was carried out as follows. First, for F = 0 the equation P(K1,K2) = 0 transformed to P(K1) = 0. From the smallest root K10 of this equation the critical weight FGcr of the column, which had not been loaded by the axial compressive force F, was determined with (47). Then, for several ratia FG/FGcr, which deter-mined the value of FG and thus the value of K1, the equation P(K1, K2) = 0 transformed to particular equa-tions P(K2) = 0. From the smallest root K20 of these equa-tions the critical forces F’cr which, in combination with the given FG would cause buckling, were deter-mined through (48). In this way, for FG = 0,Fcr0 was determined as the critical compressive force acting on the column of negligible weight. The values of FGcr, Fcr0 and F’cr were deter-mined for the following columns with geometries de-fined by n and p : 1. 2. 3. n = 4, p = 2 (the column in the form of a truncated cone or pyramid resp. hollow truncated cone or pyramid); n = 3, p = 1 (this is referring to the column with a constant width and linearly varying height of the cross-section or approximately to the column in the form of a conical tube of constant wall thickness); n = 2, p = 0 (this is approximately referring to the column constructed from four rods of constant cross-section laid along the edges of a virtual truncated pyramid connected by the lattice filling of negligible weight). The quantity dG plotted as a function of the ratio I2/I1 (Fig.1) and for the values of n and p as men-tioned above is shown in Diagram 1. The critical intrinsic weight of the column FGcr is expressed by dG as follows: EI2 d l2 u 1 t PlagpBvn 1 X {¦¦'..,1 L.4 -i ¦V, c*s.i:.. ¦;¦;¦ "i ¦ , l-r- v^ \ *-|_-'- ¦>—1_ ¦i- —f— : i T. m Ml • * ™ ¦ i ¦ (49). J. Saucha, J. Rado{, ^. Ivakovi}: Stabilnost antenskega stebra - The Stability of Antenna Column Veličina d je narisana v odvisnosti od razmerja///1 ter za omenjene vrednosti n v diagramu 2. Kritična sila F , ki deluje na steber zanemarljive teže, je povezana z dF kot: The quantity dF plotted as a function of the ratio I2/I1 and for the values of n mentioned above is shown in Diagram 2. The critical force Fcr0 acting on the column of negligible weight is expressed by dF as follows: EI2 d F2 (50). 3 PRIMERJAVA S PRIBLIŽNO REŠITVIJO Približna rešitev problema s predpostavljeno linearno povezanostjo teže stebra in vzdolžne tlačne sile F je prikazana na sliki 2 z Dunkelreyevo premico (črtkana črta) F' kjer je F’ „ vzdolžna tlačna sila, ki bi skupaj s podkritično težo stebra FG, povzročila uklon. Po postopku, ki je opisan v tem prispevku, so bile za ista razmerja FG/FG dobljene večje vrednosti F’ . Te vrednosti so podane kot krivulja na sliki 2, ki se razlikuje od Dunkelreyeve premice za razliko D. Zato lahko zapišemo FL Razmerje: 3 COMPARISON WITH THE APPROXIMATE SOLUTION The approximate solution to the problem assuming a linear interaction of the weight of the column and the axial compressive force F, is shown in Fig.2 by Dunkelrey’s straight line (dashed line) =1 (51), cr where F’crD is the axial compressive force that would, in a linear interaction with a subcritical weight FG of the column, cause buckling. With the procedure described in this paper for the same ratia FG/FGcr larger values of F’cr have been determined. These values are shown by the curve in Fig.2. This curve differs from Dunkelrey’s line by a difference D. Thus it can be written that: =1+ D (52). The ratio: FL/F F' /F crD cr 1- FG / FGcr (53), grin^sfcflMISDSD VH^tTPsDDIK stran 12 J. Saucha, J. Rado{, ^. Ivakovi}: Stabilnost antenskega stebra - The Stability of Antenna Column Sl.2. Povezanost teže stebra in vzdolžne tlačne sile Fig. 2 Interaction of the weight of the column and the axial compressive force ki ga izberemo kot merilo razlike med F’ in F’ D, narašča za večja razmerja I2/I1 in FG/FGc cr ter večje vrednosti eksponentov n in p. Največje vrednosti f presegajo 1,3 (tj. F’ prek 30 odstotkov nad F’ D). Toda, če upoštevamo razmerje kritične obremenitve (F+FG) , kakor ga daje rešitev ravnovesne enačbe, opisane v tem prispevku, ter celotna kritična obremenitev (F+FG) D približne rešitve, je največja vrednost razmerja cr pod 1,02, tj. celotna dejanska kritična obremenitev se od približne rešitve razlikuje le za +2 odstotka. Vzrok za tako majhno razliko je v dejstvu, da je FGc 3 do 10-krat večja od F 0, tako da je vpliv večje dejanske kritične sile F’ zanemarljiv v primerjavi z veliko večjo vrednostjo FG, tembolj ker največja različnost med F’ D in F’ nastaja v območju velikih razmerij F G/ f" ko je vrednost FG veliko večja od vrednosti f Gcr, 4 SKLEP To poročilo ugotavlja, da približna rešitev, ki predpostavlja linearno povezanost teže stebra in vzdolžne tlačne sile, daje celotno kritično obremenitev (F+F) D, ki se razlikuje od dejanske celotne kritične sile (F r +F) znotraj -2 odstotka in zato približna rešitev ponuja rezultate velike natančnosti in na varni strani. Zato je povečevalni količnik: which has been taken as a measure of differing F’cr from F’crD was growing as the ratia I2/I1 and FG/FGcr were larger and the exponents n and p were larger. The highest values of f are over 1.3 (i.e., F’cr over 30 % more than F’crD). But if the ratio of the total critical load (F+FG)cr determined by the solution to the equilibrium equation described in this paper to the total critical load (F+FG)crD using the approximate solution is considered, a maximum obtained value of this ratio extends under 1.02, i.e., the total real critical load differs from the one obtained by the approximate solution by less than +2 %. The reason for such a small difference is the fact that FGcr is 3 to 10 times bigger than Fcr0, and so the influence of the larger real critical force F’cr vanishes in the addition to the much larger value of FG, the more so as the largest differing of F’crD from F’cr takes place in the area of the big ratia FG/FGcr when the value of FG is much larger than the value of F’cr. 4 CONCLUSION This paper establishes that the approximate solution, assuming a linear interaction of the weight of the column and the axial compressive force, gives the total critical load (FG+F)crD which differs from the real total critical load (FG+F)cr within -2 % and therefore the approximate solution offers great accuracy and is on the side of safety. Thus the amplification factor: 1 zelo dobro približno podan z [6]: kjer je: (FG + F) (FG + F)cr can be very well approximated by [6]: 1 (54) (FG + F) (55), (FG + F)crD where: (FG+F)c 1+ F +F Gcr cr0 (56). F +F Gcr cr isfFIsJBJbJJIMlSlCšD I stran 13 glTMDDC J. Saucha, J. Rado{, ^. Ivakovi}: Stabilnost antenskega stebra - The Stability of Antenna Column Tu je F/FG dejansko razmerje delujoče obremenitve v razmerju s težo antenskega stebra. Vrednosti FGc in F 0 lahko določimmo po (49) in (50). Here F/FG is the real ratio of the effective load to the weight of the antenna column. The values of FGcr and Fcr0 can be determined by (49) and (50). 5 LITERATURA 5 REFERENCES [1] Bažant, Z.P., L. Cedolin (1991) Stability of structures. Oxford University Press, New York. [2] Chen, W.T., E.M. Lui (1987) Structural stability. Elsevier, New York/Amsterdam/London. [3] Falk, S. (1974) Knicken gerader Stäbe unter Eigengewicht. Ing.Archiv, 43:110-117. [4] Gere, M., W. Carter (1962) Critical buckling load for tapered columns. Proc. ASCE, J.Struct. Div., 88:1-11. [5] Glück, J. (1973) The buckling load of an elastically supported cantilevered column with continuously varying cross section and distributed axial load. Ing.Archiv, 42:355-359. [6] Petersen, C. (1982) Statik and Stabilität der Baukonstruktionen. FVieweg und Sohn, Braunschweig. [7] Timoshenko, S.P., J.M. Gere (1961) Theory of elastic stability. McGraw-Hill, New York Naslova avtorjev: dr. Juraj Saucha Fakulteta za strojništvo in pomorstvo I. Lučiča 5 10000 Zagreb, Hrvaška dr. Jerko Radoš dr.Čedomir Ivakovič Fakulteta za promet in prometno tehniko Vukeličeva 4 10000 Zagreb, Hrvaška Authors’ Addresses: Dr. Juraj Saucha Faculty of Mechanical Eng. and Naval Architecture I. Lučiča 5 10000 Zagreb, Croatia Dr. Jerko Radoš Dr.Čedomir Ivakovič Faculty of Transport and Traffic Engineering Vukeličeva 4 10000 Zagreb, Croatia Prejeto: Received: 6.4.2000 Sprejeto: Accepted: 12.4.2001 grin^SfcflMISDSD VH^tTPsDDIK stran 14